which is obtained by neglecting the streamwise derivatives in (REF ), and assuming the mean flow to be spatially periodic. Solving (REF ) is equivalent to performing a spatially periodic linear stability analysis (SPLSA) [1]} truncated to the number of harmonics considered in the problem. This means that the initial solution will share the same characteristics from SPLSA, including the periodicity of the solution with regards to the shock-cell wavenumber: if \(\alpha _p\)  is a complex eigenvalue of the problem, eigenvalues following \(\alpha _p \pm n\alpha _s\)  (with \(n\)  an integer) will also be solutions. The resulting eigenfunction will not be locally parallel, but spatially periodic; it may now have energy at modulation wavenumbers \(\text{Re}[\alpha _p \pm n\alpha _s]\) . This repetition of the spectrum leads to the appearance of upstream modes close to the KH mode in the spectrum, allowing for an interaction between the modes if the shock-cell amplitude is non-zero. This interaction underpins the appearance of screech tones in the flow. For the present case, this interaction is undesirable, as PFE is unable to account for upstream-travelling waves in its original form. The Kelvin-Helmholtz eigenmode is selected as the initial perturbation for all cases studied herein.
